TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000–000
S 0002-9947(XX)0000-0
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
YUKIO KASAHARA AND NICHOLAS H. BINGHAM
Abstract. We are concerned with a rather unfamiliar condition in the theory
of the orthogonal polynomials on the unit circle. In general, the Szegö function
is determined by its modulus, while the condition in question is that it is also
determined by its argument, or in terms of the function theory, that the square
of the Szegö function is rigid. In prediction theory, this is known as a spectral
characterization of complete nondeterminacy for stationary processes, studied
by Bloomfield, Jewel and Hayashi (1983) going back to a small but important
result in the work of Levinson and McKean (1964). It is also related with the
cerebrated result of Adamyan, Arov and Krein (1968) for the Nehari problem,
and there is a one-one correspondence between the Verblunsky coefficients
and the negatively indexed Fourier coefficients of the phase factor of the Szegö
function, which we call a Nehari sequence. We presents some fundamental
results on the correspondence, including extensions of the strong Szegö and
Baxter’s theorems.
1. Introduction
Let µ be a probability measure on the unit circle T = {|z| = 1} parameterized
by z = eiθ . It is called trivial if it is supported on a finite set. In the other
case, so nontrivial , µ generates a system of orthogonal polynomials on the unit
circle (OPUC) obtained by the Gram–Schmidt method applied to 1, z, z 2 , . . . in the
Hilbert space L2 (µ). Let Φ0 , Φ1 , Φ2 , . . . be the monic OPUC, and put an = Φn (0)
for n = 1, 2, . . .. These constants appear in the Szegö recurrence formula [Sz1]
(1.1)
Φn+1 = zΦn + an+1 z n Φn ,
and form a sequence a = (a1 , a2 , . . .) on the open unit disc D = {|z| < 1}. Every
sequence on D arises in this way from a unique nontrivial probability measure on T.
The last fact was first proved by Verblunsky [V1, V2] in a slightly different context,
and an take his name, the Verblunsky coefficients of µ (also called Schur, Szegö or
Geronimus coefficients or parameters). It is a central question in the spectral theory
of OPUC how the properties of a correspond to the properties of µ and vice-versa.
The strong Szegö theorem [Sz2] and Baxter’s theorem [B] to be discussed later are
cerebrated results in this direction. For OPUC, see Simon [Si1, Si2, Si3].
Let m be the normalized Lebesgue measure on T. For 1 ≤ p ≤ ∞, denote by Lp
the Lebesgue space on T with respect to m, and by H p the allied Hardy space on D.
As usual, a function in H p will be identified with its boundary value function in Lp .
For a textbook account of Hardy functions, see Hoffman [Ho]. A nonzero function g
2000 Mathematics Subject Classification. Primary 42C05; Secondary 42A10, 42A70.
Key words and phrases. Orthgonal polynomials on the unit circle, Verblunky coefficients, Nehari problem, rigid functions.
c
°XXXX
American Mathematical Society
1
2
Y. KASAHARA AND N. H. BINGHAM
in H 1 is called rigid (or strongly outer ) if it is determined by its argument, or more
specifically, by the unimodular function g/|g| up to a positive constant factor. A
rigid function is outer, but the converse is not true. For rigidity and related topics,
see Sarason [S3] and Poltoratski–Sarason [PS]. Rigidity arises as key ingredients
in many problems, some of which will be discussed soon below and later, while it
does not seem to have received considerable attention in OPUC. One of the chief
aims of this paper is to elaborate its role in OPUC.
Write dµ = wdm + dµs , where µs is the singular part. When log w ∈ L1 , there
is a unique outer function h in H 2 such that h(0) > 0 and w = |h|2 a.e. on T.
It is called the Szegö function, and plays a significant role in the spectral theory
of OPUC. Note that h2 is an outer function in H 1 with h2 /|h2 | = h/h̄. In their
investigation for the dependence structure of continuous-time stationary Gaussian
processes in terms of weighted L2 -space on the line, Levinson–McKean [LM] found
a spectral density relevant to rigidity. For its analogue on the circle (known as a
characterization of complete nondeterminacy for discrete-time stationary processes,
studied by Bloomfield et al. [BJH]), let us call w a Levinson–McKean weight (for
short, LM-weight) if
(1.2)
µs = 0,
log w ∈ L1 ,
and
h2 is rigid.
See also Dym–McKean [DM] for background. Under the condition (1.2), the space
L2 (µ) has a certain structure that permits us to handle OPUC Φn in a particular
manner (using von Neumann’s alternating projections theorem) of the kind treated
by Seghier [Se], Inoue [In1, In2, In3], Inoue–Kasahara [IK1, IK2], and Bingham et
al. [BIK] in the context of prediction theory. In [LM], the key phrase in (1.2) is
actually described as “h is determined by its phase factor h/h̄ ”, and this means
that µ is determined by it, or equivalently, by the two-sided sequence of its Fourier
coefficients. In fact, rigidity enables us to specify µ only from the negative half of
the sequence, as explained now.
Rigidity also appears in the so-called Nehari problem, in which, given a sequence
γ = (γ1 , γ2 , . . .) Rof complex numbers, one seeks functions φ in the unit ball of L∞
such that γn = T einθ φ dm for all n = 1, 2, . . .. The name came from Nehari [Ne]
proving that a solution exists if and only if the Hankel matrix made up from γ acts
as a contraction on the space `2 . The problem has more than one solution if and
only if γ consists of the negatively indexed Fourier coefficients for the phase factor of
some function in H 2 . In this indeterminate case, γ will be called a Nehari sequence.
Adamyan–Arov–Krein [AAK] described the solution set of an indeterminate Nehari
problem using an outer function h in H 2 with the following properties: h has unit
norm, h2 is rigid, and h/h̄ solves the problem, namely, for n = 1, 2, . . .,
Z
(1.3)
γn =
einθ (h/h̄)dm.
T
Moreover, h can be chosen so h(0) > 0. Then h is uniquely determined by γ, and
this relation sets up a one-one correspondence between the set of Nehari sequences
and the set of LM-weights. For textbook treatments of the Nehari problem, see
Peller [P] (operator-theoretic approach originating with [AAK]) and Garnett [G]
(function-theoretic approach).
To sum up the above discussion involved with (1.1)–(1.3), a probability measure
µ with LM-weight has two kinds of parameters, its Verblunsky coefficients a and its
Nehari sequence γ. Thus, besides the ‘central question’ mentioned earlier, it is also
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
3
of interest to study its analogue for a and γ. The main purpose of this paper is to
present some fundamental results for these questions. The outline is as follows. The
next section is devoted to a review of background information on Hardy functions
and the concept of rigidity. In Section 3, our research starts with a discussion
on the structure of L2 (µ), stemming from a discovery of Levison–McKean, and
related matters in OPUC. Section 4 proceeds to the Adamyan–Arov–Krein theory
for establishing some basic relations between a and γ. The final section provides
the strong Szegö and Baxter’s theorems in terms of a and γ.
2. Rigidity
In this preparatory section, we recall some basic matters in the theory of Hardy
functions and the concept of rigidity.
For 1 ≤ p ≤ ∞, let Lp be the Lebesgue space with respect to the normalized
Lebesgue measure m on the unit circle T, and write k · kp for the usual Lp -norm.
The Hardy space H p is a Banach space of analytic functions f (z) on the unit disc D
such that supr<1 kfr kp < ∞, the supremum being the norm. Here, fr (eiθ ) = f (reiθ )
and its Lp -norm is nondecreasing in r < 1, so that the supremum coincides with the
limit as r → 1. An H p -function f (z) has its boundary-value function f = limr→1 fr
in Lp , from which the original function can be recovered by the Poisson integral. In
p
what follows, these two functions will not be distinguished,
R inθ and H will be regarded
p
p
p
as a closed subspace of L , so that H = {f ∈ L | T e f dm = 0 for n = 1, 2, . . .}.
Note that H p ⊃ H q for 1 ≤ p < q ≤ ∞. A nonzero function f ∈ H 1 satisfies
log|f | ∈ L1 , and so it has positive modulus, namely, |f | > 0 a.e.. A function
j ∈ H ∞ is called inner if |j| = 1 a.e.. By the maximum principle, |j(z)| < 1 on D
unless j is constant. A nonzero function g ∈ H 1 is called outer if
µZ iθ
¶
e +z
iθ
(2.1)
g(z) = c · exp
log|g(e
)|dm
(z ∈ D),
iθ
T e −z
in which c = g(0)/|g(0)|, or equivalently, if it has the following extremal property:
if f ∈ H 1 satisfies |f | ≤ |g| a.e., then |f (z)| ≤ |g(z)| on D. Every nonzero function
f ∈ H 1 is expressed as f = jg with inner j and outer g satisfying |f | = |g| a.e.,
and this inner-outer factorization is unique up to constant factors of modulus one.
In a Banach space, a point of the unit ball B is called an exposed point of B if
there exists a linear functional on the space that attains its norm at the point and
only there, and it is necessarily an extreme point of B, that is, a point in B which
is not a proper convex combination of two distinct points in B. As is well-known,
de Leeuw–Rudin [dLR] showed that an H 1 -function is an extreme point of the unit
ball of H 1 if and only if it is an outer function with unit norm. They also discussed
the exposed points of the ball. A linear functional L on H 1 that
R has g in the unit
ball of H 1 attaining L(g) = kLk can be expresses as L(f ) = T φf dm (f ∈ H 1 )
with φ = kLk · |g|/g, and then L(f ) = kLk implies f /|f | = g/|g| a.e. as well as
kf k1 = 1. Hence, the exposed point in question is an H 1 -function with unit norm
for which no other functions in the unit sphere of H 1 have the same argument.
This gives rise to the concept of rigidity to be introduced now.
A nonzero function g ∈ H 1 is called rigid (Sarason [S2]) if it is determined by
its argument up to a positive constant factor, or more precisely, if f /|f | = g/|g|
a.e. implies f = cg for some c > 0, subject to f ∈ H 1 . Such g is also called
strongly outer (Nakazi [N1]). It is a set routine to compare two function f = jo
4
Y. KASAHARA AND N. H. BINGHAM
and g = (1 + j)2 o with inner j and outer o. Note that g is outer because 1 + j is
so (see [Ho, p.76]). In view of (1 + j)2 = j|1 + j|2 , they enjoy f /|f | = g/|g|, but
g/f = |1 + j|2 is non-constant unless j is constant. This shows on one hand that
non-outer functions cannot be rigid, and on the other hand that there are nonrigid
outer functions. Unfortunately, no one has found a structural characterization of
rigid functions, and there is no good answer to the question which outer functions
are rigid. Each of the following two conditions is sufficient for a function in H 1 to
be rigid. One is that its reciprocal is also in H 1 , and the other is that its real part
is nonnegative (see Yabuta [Y]). See Sarason [S3] and Poltoratski–Sarason [PS] for
further information on rigidity.
3. Levinson–McKean weights
In this section, we discuss the structure of L2 (µ) with LM-weight and relevant
matters, including a useful expansion formula for OPUC.
2
Let µ be a probability measure on T. Denote by H 2 (µ) and H−
(µ) the closed
2
2
2
subspaces of L (µ) spanned by 1, z, z , . . . and z̄, z̄ , . . ., respectively, and by H 2
2
and H−
those spaces for the normalized Lebesgue measure m. This kind of set-up
is often used for studying a discrete-time, wide-sense, stationary stochastic process
2
(µ) are interpreted as its past and
with spectral measure µ, and H 2 (µ) and H−
2
future. The process is called completely nondeterministic if H−
(µ) ∩ H 2 (µ) = {0},
according to Sarason [S1]. Recall dµ = wdm + dµs , where µs is the singular part.
The Szegö alternative asserts that
L2 (µs ) = ∩n [z n H 2 (µ)] ( H 2 (µ) ( L2 (µ)
if log w ∈ L1 , otherwise ∩n [z n H 2 (µ)] = H 2 (µ) = L2 (µ) (cf. Dym–McKean [DM]).
This shows that µs = 0 and log w ∈ L1 are necessary for complete nondeterminacy.
If log w ∈ L1 , there is a unique outer function h ∈ H 2 satisfying w = |h|2 a.e. and
h(0) > 0. It is called the Szegö function, and given by
µZ iθ
¶
√
e +z
h(z) = exp
(z ∈ D).
log w dm
iθ
T e −z
For n = 1, 2, . . ., let Pn be the space of polynomials with degree less than n, and
put P0 = {0}. Let us now recall the definition (1.2) of an LM-weight. It originated
with Levinson–KcKean [LM, p. 105], in which they found (1.2) to be a spectral
characterization of the identity
(3.1)
2
z n H−
(µ) ∩ H 2 (µ) = Pn
for n = 1 (actually, this is an analogue of their result in the weighted L2 -space
on the real line, but it can be proved in just the same way as they did). It seems
that their discovery had not received considerable attention until Bloomfield et
al. [BJH, Proposition 5] characterized complete nondeterminacy in terms of the
exposed points in the unit ball of H 1 . It is easy to see that the occurrence of (3.1)
does not depend on n = 0, 1, 2, . . ., as long as µ is nontrivial. Indeed, one has
2
2
2
z n−1 H−
(µ) ∩ H 2 (µ) = [z n H−
(µ) ∩ H 2 (µ)] ∩ z̄ [z n H−
(µ) ∩ H 2 (µ)],
2
2
while if log w ∈ L1 , it follows from z n+1 H−
(µ) = z n H−
(µ) + z n C that
2
2
z n+1 H−
(µ) ∩ H 2 (µ) = [z n H−
(µ) ∩ H 2 (µ)] + z n C.
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
5
So, by induction, (3.1) holds for every n = 0, 1, 2, . . . if it holds for some n (cf. Inoue–
Kasahara [IK2, Theorem 2.3]), except for the trivial case that µ is supported on a
finite set, which makes Pn = L2 (µ) for all n ≥ ] supp(µ).
The following is a consequence of the above discussion on the structure of L2 (µ)
with LM-weight (cf. [IK2, Remark 2]).
Theorem 3.1. Let µ be a probability measure on T. If µ satisfies (1.2), then (3.1)
holds for every n = 0, 1, 2, . . .. Conversely, if µ is nontrivial and (3.1) holds for
some n = 0, 1, 2, . . ., then µ satisfies (1.2).
Remark 3.2. To the best of our knowledge, the full statement of Theorem 3.1 has
never appeared before, while the essence of the theorem was clarified in the 1980s.
For instance, the point of the matter is implicitly given in [BJH, Propositions 9]
as a spectral characterization of k-step complete nondeterminacy. Some relevant
results may also be found in Hayashi [Ha, Theorem 1], Nakazi [N2, Theorem 9] and
Younis [Yo, Theorem 4].
Let us turn to the monic OPUC Φn , which are the projections of z n onto Pn⊥
in L2 (µ). If µs = 0 and log w ∈ L1 , the mapping f → hf from L2 (µ) to L2 is a
Hilbert space isomorphism, and Beurling’s theorem (see [Ho, p.101]) implies
2
2
h[z n H−
(µ) ∩ H 2 (µ)] = z n (h/h̄)H−
∩ H 2.
This is quite useful in studying the OPUC Φn for an LM-weight. Indeed, by (3.1),
the image hΦn is obtained by projecting z n h onto the orthogonal complement of
2
z n (h/h̄)H−
∩H 2 = hPn in L2 , and this kind of projection can be described in terms
of the Toeplitz and Hankel operators, as we now explain.
Let φ ∈ L∞ , and write Mφ for the multiplication operator on L2 defined by
Mφ f = φf . The Toeplitz operator Tφ on H 2 and the Hankel operator Hφ from H 2
2
to H−
with symbol φ are defined by
Tφ = P Mφ |H 2 ,
Hφ = P− Mφ |H 2 ,
2
where P and P− are the orthogonal projection operators of L2 onto H 2 and H−
,
respectively. Clearly, Tφ + Hφ = Mφ |H 2 . Their adjoint operators are given by
Tφ∗ = P Mφ̄ |H 2 = Tφ̄ ,
Hφ∗ = P Mφ̄ |H−2 .
The projection just stated can be treated in the following approximation scheme.
Lemma 3.3. Let φ be a unimodular function in L∞ , and let Qφ be the orthogonal
2
projection operator of H 2 onto φH−
∩ H 2 . Then, for f ∈ H 2 , it holds that
f − Qφ f =
∞
X
Tφ [Hφ∗ Hφ ]k Tφ∗ f,
k=0
the sum converging strongly in H 2 .
Proof. Let |φ| = 1. Then Mφ P− Mφ̄ is the orthogonal projection operator of L2 onto
2
φH−
. Since Hφ̄∗ Hφ̄ = P Mφ P− Mφ̄ |H 2 , it follows from von Neumann’s alternating
projections theorem (see Halmos [H, Problem 122]) that (Hφ̄∗ Hφ̄ )n converges to Qφ
as n → ∞ in the strong operator topology. Recall that
I − Hφ̄∗ Hφ̄ = Tφ Tφ∗ ,
Hφ̄∗ Hφ̄ Tφ = Tφ Hφ∗ Hφ
6
Y. KASAHARA AND N. H. BINGHAM
(see Peller [P, (3.1.5) and (4.4.1)]). Repeated use of these relations gives
I−
(Hφ̄∗ Hφ̄ )n
=
n−1
X
Tφ [Hφ∗ Hφ ]k Tφ∗ ,
k=0
whence the lemma follows.
¤
I−Hφ∗ Hφ
Remark 3.4. If
is invertible, the above result may be rewritten as Seghier’s
formula [Se, Proposition 3], namely,
f − Qφ f = φ(I − Hφ∗ Hφ )−1 Tφ∗ f − Hφ (I − Hφ∗ Hφ )−1 Tφ∗ f.
For convenience, write Tn and Hn for the Toeplitz and Hankel operators with
symbol φ = z n (h/h̄). By Theorem 3.1, when µ has an LM-weight, its monic OPUC
admit the following expansion.
Proposition 3.5. If µ satisfies (1.2), then, for every n = 0, 1, 2, . . .,
∞
X
hΦn = h(0)
Tn [Hn∗ Hn ]k 1,
k=0
the sum converging strongly in H 2 .
Proof. Apply Lemma 3.3 to f = z n h, noting Tn∗ (z n h) = P h̄ = h(0).
¤
The above is a consequence from Theorem 3.1 that if µ has an LM-weight,
(3.2)
2
Φn is perpendicular to z n H−
(µ) ∩ H 2 (µ).
To see the converse implication, recall Hayashi’s criterion for rigidity [Ha, p.695]:
2
∩ H 2,
h2 is rigid if and only if h is perpendicular to (h/h̄)H−
provided that h is a nonzero function in H 2 . This can be extended to the following
characterization of LM-weights in terms of OPUC.
Theorem 3.6. Let µ be a probability measure on T. If µ satisfies (1.2), then (3.2)
holds for every n = 0, 1, 2, . . .. Conversely, if µ is nontrivial and (3.2) holds for
some n = 0, 1, 2, . . ., then µ satisfies (1.2).
Proof. The converse half is to be checked. Since µ is nontrivial, Φn never vanishes,
and (3.2) has a proper meaning. If it holds for some n = 1, 2, . . . , by symmetry,
2
z n Φn is perpendicular to z[z n H−
(µ) ∩ H 2 (µ)], so that Φn−1 is perpendicular to
n−1 2
2
z
H− (µ) ∩ H (µ) in view of the inverse Szegö recursion
n
zΦn−1 = ρ−2
n [Φn − an z Φn ],
p
where ρn = 1 − |an |2 , and thus, by induction, (3.2) holds for n = 0. Therefore,
2
in any case, the assumption implies that 1 is perpendicular to H−
(µ) ∩ H 2 (µ). The
2
Szegö alternative now allows us to have µ = |h| m, showing that h is perpendicular
2
to (h/h̄)H−
∩ H 2 . Consequently, by Hayashi’s criterion, µ has an LM-weight. ¤
Remark 3.7. It is also interesting to see Hayashi’s criterion through our approxima2
tion scheme. Take a nonzero function h in H 2 and project it onto (h/h̄)H−
∩ H 2.
∗
Then, since Th/h̄ h = h(0), it follows from Lemma 3.3 that
kh − Qh/h̄ hk22 = (h − Qh/h̄ h, h)2 = |h(0)|2
∞
X
∗
k
([Hh/
h̄ Hh/h̄ ] 1, 1)2 ,
k=0
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
7
which, by Hayashi’s criterion, coincides with khk22 if and only if h2 is rigid. This
should be compared with [Ha, Theorem 8], which states that the equality holds in
|h(0)|/ inf f ∈H 2 kTh/h̄ (1 + zf )k2 ≤ khk2 if and only if h2 is rigid.
4. Verblunsky coefficients and Nehari sequences
In this section, after recalling the result of Adamyan–Arov–Krein, we study how
a Nehari sequence corresponds, via an LM-weight, to its Verblunsky coefficients.
By Herglotz’s theorem, the association
Z iθ
1 + zf (z)
e +z
(4.1)
=
dµ
(z ∈ D)
iθ
1 − zf (z)
T e −z
sets up a one-one correspondence f ↔ µ between the unit ball of H ∞ and the set
of probability measures on T. Here, f is called the Schur function of µ, and the
latter is trivial if and only if f is a finite Blaschke product. Also, since
(4.2)
w=
1 − |f |2
|1 − zf |2
a.e.,
where w is the density in dµ = wdm + dµs , the Szegö condition log w ∈ L1 is
fulfilled if and only if log(1 − |f |) ∈ L1 , which means that f is not an extreme point
of the unit ball of H ∞ (see [Ho, p.138]). If f is not a finite Blaschke product, first
putting f0 = f and then iterating
(4.3)
αn = fn (0),
fn+1 (z) =
1 fn (z) − αn
,
z 1 − ᾱn fn (z)
one has a sequence of functions f0 , f1 , f2 , . . ., called the Schur iterates of f , in the
unit ball of H ∞ . The numbers αn are called the Schur parameters of f . In fact,
they are essentially the same as the Verblunsky coefficients an for µ, namely,
an = −ᾱn−1 .
This fact is Geronimus’ theorem. See Simon [Si1] for background.
Let γ = (γ1 , γ2 , . . .) be a sequence of complex numbers. In the Nehari problem,
∞
one seeks functions φ in the unit ball
R of L that have γ as its negatively indexed
Fourier coefficients, that is, γn = T einθ φdm for all n = 1, 2, . . .. In this paper, as
mentioned earlier, γ is called a Nehari sequence if the problem has more than one
solution. Adamyan–Arov–Krein [AAK] showed that such γ has an outer function
h with unit norm that yields a bijection u → φ from the unit ball of H ∞ onto the
solution set, defined by the formula
(4.4)
φ = (h/h̄) −
h2 (1 − zf )(1 − u)
,
1 − zf u
where f is the Schur function of µ = |h|2 m. In particular, h2 is rigid, and this is
essential for describing the full solutions; as long as h ∈ H 2 satisfies (1.3) and has
unit norm, the formula (4.4) provides many solutions as u runs through the unit
ball of H ∞ . The above formula may be rewritten as
φ=
h(1 − zf )(u − zf )
.
h̄(1 − zf )(1 − zf u)
So, |φ| = 1 a.e. if and only if u is inner, in view of
|1 − zf u|2 − |u − zf |2 = (1 − |f |2 )(1 − |u|2 ).
8
Y. KASAHARA AND N. H. BINGHAM
For λ ∈ T, the solution φ corresponding to u = λ is called canonical . It arises as
the phase factor φ = hλ /h̄λ of the function
√ h(1 − zf )
hλ = λ
,
1 − z(λf )
√
in which λ stands for one of the square-roots of λ. This hλ has unit norm, and
its square is rigid. In particular, it plays the same role as h in describing the
full solution set. In this connection, the measures µλ = |hλ |2 m, associated with
fλ = λf , constitute the family of the Alexandrov–Clark measures of µ = |h|2 m (see
Poltoratski–Sarason [PS] and the references cited there).
Now,
√ in the above result of [AAK], one may assume h(0) > 0 (otherwise take hλ
with λ = |h(0)|/h(0) instead of h). Then γ is associated, via the Szegö function
h satisfying (1.3), with a probability measure µ = |h|2 m with LM-weight. This
defines a one-one correspondence between the set of Nehari sequences and the set
of LM-weights. Moreover, by Verblunsky’s theorem, a Nehari sequence γ is also
associated with a unique Verblunsky coefficients a in such a way that
(4.5)
a = (a1 , a2 , . . .)
↔
µ
↔
γ = (γ1 , γ2 , . . .).
In view of this, the question naturally arises as to how a Nehari sequence corresponds, via an LM-weight, to its Verblunsky coefficients.
The main aim here is to establish the following commutativity between the correspondence a ↔ γ and the unilateral shift (x1 , x2 , . . .) → (x2 , x3 , . . .), and the
‘inverse’ relation. (These are relevant to the ‘coefficient stripping’ in [Si1].)
Theorem 4.1. For a = (a1 , a2 , . . .) corresponding to γ = (γ1 , γ2 , . . .) via µ satisfying (1.2) as in (4.5), the following hold:
(i) ã = (a2 , a3 , . . .) corresponds to γ̃ = (γ2 , γ3 , . . .), via a certain probability
measure µ̃ satisfying (1.2).
(ii) For any ζ ∈ D, aζ = (ζ, a1 , a2 , . . .) corresponds, via a certain probability
measure µζ satisfying (1.2), to γζ = (ωζ , γ1 , γ2 , . . .) with
Z
(4.6)
ωζ = (h/h̄) dm − h(0)2 (1 − ζ),
T
where h is the Szegö function of µ.
Before passing to the proof, let us discuss a pair of immediate by-products of
the above theorem. In OPUC, one expects that the substantial properties of µ are
determined by the asymptotic behavior of a. The following indicates that this also
applies to LM-weights. (Compare [Si1, Theorem 3.4.4].)
Corollary 4.2. Let µ and ν be two nontrivial probability measures on T with
Verblunsky coefficients a and b, respectively. Suppose for some K and k > −K,
an = bn+k
(n ≥ K).
Then either both µ and ν satisfy (1.2), or neither does.
This can be rephrased in terms of the Schur functions, as follows.
Corollary 4.3. Let f0 , f1 , f2 , . . . be the Schur iterates arising from a nontrivial
probability measure. Then either all of them correspond, as the Schur functions, to
probability measures obeying (1.2), or none of them do.
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
9
Theorem 4.1 (i) readily follows from the next expansion formula for the Verblunsky coefficients for an LM-weight. Recall that Tn and Hn stand for the Toeplitz
and Hankel operators with symbol z n (h/h̄).
Proposition 4.4. Let µ be a probability measure on T. If it satisfies (1.2), then
an =
∞
X
(Tn [Hn∗ Hn ]k 1, 1)2
k=0
holds for every n = 1, 2, . . ..
Proof. Since an = Φn (0), we have hΦn ≡ an h (mod zH 2 ). By Proposition 3.5,
an h(0) = (an h, 1)2 = (hΦn , 1)2 = h(0)
∞
X
(Tn [Hn∗ Hn ]k 1, 1)2 .
k=0
Thus, the assertion follows.
¤
Proof of Theorem 4.1 (i). Let a, µ and γ be as in (4.5). Then γ̃ = (γ2 , γ3 , . . . )
is a Nehari sequence having a solution φ = z(h/h̄), and it corresponds to some
probability measure µ̃ = |h̃|2 m satisfying (1.2). Write T̃n and H̃n for the Toeplitz
¯
and Hankel operators with symbol z n (h̃/h̃). Since the phase factor of h̃ solves the
Nehari problem for γ̃, it follows that
∗
T̃n∗ 1 = Tn+1
1.
H̃n = Hn+1 ,
So, by Proposition 4.4, the Verblunsky coefficients ãn of µ̃ satisfy
ãn =
∞
X
(T̃n [H̃n∗ H̃n ]k 1, 1)2
=
∞
X
∗
(Tn+1 [Hn+1
Hn+1 ]k 1, 1)2 = an+1 ,
k=0
k=0
which means ã = (a2 , a3 , . . . ).
¤
Remark 4.5. By Proposition 4.4, the map γ → a from the set of Nehari sequences
to the set of their Verblunsky coefficients may be visualized as
an =
∞
X
t
γ n (Γ∗n Γn )k e,
γn+2
γn+3
γn+4
γn+3
γn+4
γn+5

···
··· 
,
··· 


1
 0 

e=
 0 .
...
γn+1
 γn+2
Γn = 
 γn+3
...

.
...

γn
 γn+1 

γn = 
 γn+2  ,
...

..
where
...
k=0
(From this, it is easily seen that λγ, which is a Nehari sequence, corresponds to λa
for any λ ∈ T.) In this connection, when µ has an LM-weight, one may write
kΦn k2µ = h(0)2
∞
X
t
e (Γ∗n Γn )k e.
k=0
See Inoue [In1, In2, In3], Inoue–Kasahara [IK1, IK2] and Bingham et al. [BIK] for
relevant results with application in prediction theory.
10
Y. KASAHARA AND N. H. BINGHAM
Theorem 4.1 (ii) can be derived from (i) with the aid of Geronimus’ theorem
and the next parameterization for the one-step extensions of a Nehari sequence
γ = (γ1 , γ2 , . . . ). To state it, put
D(γ) = {ωζ | ζ ∈ D} ,
where ωζ is Rthe constant displayed in (4.6). This is an open disc of radius h(0)2
centered at T (h/h̄)dm − h(0)2 . Also, it follows that D(γ) ⊂ D.
Lemma 4.6. Let γ = (γ1 , γ2 , . . . ) be a Nehari sequence. Then its one-step extension (ω, γ1 , γ2 , . . . ) remains a Nehari sequence if and only if ω ∈ D(γ).
Proof. For any ζ ∈ D, the extension (ωζ , γ1 , γ2 , . . . ) is a Nehari sequence, for
·
¸
h2 (1 − zf )(1 − u)
φ = z̄ (h/h̄) −
1 − zf u
solves its Nehari problem as long as u(0) = ζ. Every extension is of such form.
Indeed, if (ω, γ1 , γ2 , . . . ) is a Nehari sequence, the
R problem has a solution of the
form k/k̄ with k ∈ H 2 . This means that γn = T einθ z(k/k̄)dm for n = 1, 2, . . .,
namely, z(k/k̄) is a solution of the Nehari problem for the original sequence γ. So
there is an inner function u such that
z(k/k̄) = h/h̄ −
h2 (1 − zf )(1 − u)
,
1 − zf u
and the additional entry ω is evaluated as
Z
Z
iθ
ω=
e (k/k̄)dm = (h/h̄)dm − h(0)2 (1 − u(0)).
T
T
Here u(0) ∈ D because otherwise u = λ (λ = u(0) ∈ T) leads to z(k/k̄) = hλ /h̄λ ,
contradicting the fact that h2λ is rigid. So ω lies in the disc D(γ).
¤
Proof of Theorem 4.1 (ii). Let a, µ and γ be as in (4.5). Take ζ ∈ D. Then, by
Lemma 4.6, γζ = (ωζ , γ1 , γ2 , . . . ) is a Nehari sequence, and it corresponds to the
Verblunsky coefficients, say, aζ of a probability measure µζ = |hζ |2 m obeying (1.2).
By Theorem 4.1 (i), aζ is specified up its first entry, denoted by ζ0 , as
aζ = (ζ0 , a1 , a2 , . . .).
It remains to check ζ0 = ζ. Let f and fζ be the Schur functions of µ = |h|2 m and
µζ = |hζ |2 m, respectively. Geronimus’ theorem plays a crucial role here. Namely,
it implies both that fζ (0) = −ζ̄0 and that fζ has f as its first Schur iterate. The
upshot from these two facts is that
fζ (z) =
zf (z) − ζ̄0
,
1 − ζ0 zf (z)
and a direct computation involving (4.2) gives
p
1 − |ζ0 |2 h(z)(1 − zf (z))
hζ (z) =
·
1 − zf (z)bζ0 (z)
1 + ζ̄0 z
whence
z(hζ /h̄ζ ) = (h/h̄) −
with
bζ0 (z) =
h2 (1 − zf )(1 − bζ0 )
.
1 − zf bζ0
z + ζ0
,
1 + ζ̄0 z
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
11
Since bζ0 (0) = ζ0 , the proof is finished by
Z
Z
ωζ =
eiθ (hζ /h̄ζ )dm = (h/h̄)dm − h(0)2 (1 − ζ0 ),
T
T
showing ζ0 = ζ, as required.
¤
Since rigidity is a key concept of this paper, it would be worthwhile to single out
the following outcome from the above proof, together with a direct proof.
Proposition 4.7. Let h be a Szegö function. Suppose that h has unit norm and
h2 is rigid. Also, let f be the Schur function of µ = |h|2 m. Then for any ζ ∈ D,
the Szegö function hζ , defined by
p
1 − |ζ|2 h(z)(1 − zf (z))
z+ζ
hζ (z) =
·
with bζ (z) =
,
1 − zf (z)bζ (z)
1 + ζ̄z
1 + ζ̄z
inherits its predecessor’s property, viz, hζ has unit norm and h2ζ is rigid.
A direct proof of Proposition 4.7. Thanks to Sarason [S2, Theorem 2], the two
properties of hζ in question can be deduced from those of the function
gζ (z) =
h(z)(1 − zf (z))
.
1 − zf (z)bζ (z)
That is to say, since bζ is inner, Sarason’s theorem implies that gζ has unit norm
and gζ2 is rigid. Now, (4.2) gives |h(1 − zf )|2 = 1 − |f |2 , so that
|gζ |2 =
1 − |f bζ |2
.
|1 − zf bζ |2
Hence, it follows from kgζ k2 = 1 that
Z iθ
e +z
1 + zf (z)bζ (z)
|gζ |2 dm =
,
iθ
1 − zf (z)bζ (z)
T e −z
and since bζ (−ζ) = 0, the value of this function at z = −ζ is one. Therefore,
·Z iθ
¸
Z
1 − |ζ|2
e −ζ
2
2
khζ k22 =
|g
|
dm
=
Re
|g
|
dm
= 1.
ζ
ζ
iθ
iθ 2
T |1 + ζ̄e |
T e +ζ
To check rigidity, take an H 2 -function k satisfying k/k̄ = hζ /h̄ζ , which implies that
(1 + ζ̄z)k/(1 + ζ̄z)k = gζ /ḡζ . Since gζ2 is rigid, (1 + ζ̄z)k is a real multiple of gζ , so
that k is a real multiple of hζ . Thus, h2ζ is rigid.
¤
Remark 4.8. The above fact provides another proof of Theorem 4.1 as follows. Let
f be the Schur function that generates a = (a1 , a2 , . . .), and also let ζ ∈ D. By
Geronimus’ theorem, the Verblunsky coefficients aζ = (ζ, a1 , a2 , . . .) arise from
zf (z) − ζ̄
.
1 − ζzf (z)
Then Proposition 4.7 implies that the probability measure µζ corresponding to aζ
has an LM-weight dµζ = |hζ |2 dm, and it is seen from the formula
fζ (z) =
z(hζ /h̄ζ ) = (h/h̄) −
h2 (1 − zf )(1 − bζ )
1 − zf bζ
that the Nehari sequence of µζ is just as in Theorem 4.1 (ii), from which (i) readily
follows. The detail is omitted.
12
Y. KASAHARA AND N. H. BINGHAM
5. Strong Szegö and Baxter’s theorems
In this final section, we extend the Strong Szegö and Baxter’s theorems to the
correspondence between the Verblunsky coefficients and the Nehari sequences.
Let N + be the Smirnov class on D, the linear space of all quotients u/v with
u, v ∈ H ∞ , v being outer. A nonzero function f ∈ N + satisfies log|f | ∈ L1 , and
(2.1) defines the outer functions in N + as in the case of H p . So, a nonzero function
f ∈ N + admits the inner-outer factorization f = jg with inner j and outer g. The
Hardy space H p is expressed as H p = Lp ∩ N + . (This is also valid for 0 < p < 1;
in this case, H p is defined in the same way as in Section 2, but it is not a Banach
space.) These basic matters on the class N + may be found in Duren [D].
Under the restriction log w ∈ L1 for dµ = wdm + dµs , the correspondence (4.1)
between Schur functions f and probability measures µ can be modified as follows.
Proposition 5.1. The formula
σ(z) + zτ (z)
=
σ(z) − zτ (z)
(5.1)
Z
T
eiθ + z
dµ
eiθ − z
(z ∈ D)
defines a one-one correspondence τ ↔ µ between the Smirnov class N + and the set
of probability measures with the Szegö condition log w ∈ L1 , provided that σ is the
unique outer function in N + determined by τ via
σ(0) > 0,
|σ|2 − |τ |2 = 1 a.e..
Further, the Szegö function h of µ satisfies
h−1 = σ − zτ a.e.,
h = (h/h̄)σ + zτ a.e..
and consider the outer function σ ∈ N + given by
·Z iθ
¸
p
e +z
2 dm
σ(z) = exp
1
+
|τ
|
(z ∈ D).
log
iθ
T e −z
Proof. Pick τ ∈ N
+
Then the quotient f = τ /σ lies in the unit ball of H ∞ , and enjoys log(1 − |f |) ∈ L1
because of 1 − |f |2 = |σ|−2 . So, f is the Schur function of a probability measure µ
with log w ∈ L1 , and their relation (4.1) gives (5.1). The Szegö function h satisfies
h = 1/(σ − zτ ) because w = 1/|σ − zτ |2 follows from (4.2), and thus,
h̄−1 σ + h−1 zτ = |σ|2 − |τ |2 = 1,
showing h = (h/h̄)σ + zτ . The other half can be checked in a similar discussion on
the pair σ = 1/h(1 − zf ) and τ = f /h(1 − zf ) arising from f and h of µ.
¤
Remark 5.2. For 0 < p ≤ ∞, one has σ ∈ H p if and only if τ ∈ H p . (Indeed,
|τ |p ≤ |σ|p ≤ 2p/2 (|τ |p + 1) for 0 < p < ∞.) Since |h|2 = σh + zτ h, µ is absolutely
continuous if τ h ∈ H 1 (or equivalently, if σh ∈ H 1 ). In particular, µ satisfies (1.2)
if τ ∈ H 2 , which implies w−1 ∈ L1 . Actually, τ belongs to H 2 if and only if µs = 0,
log w ∈ L1 and h ∈ Th/h̄ H 2 .
Let µ be a probability measure with log w ∈ L1 , and let σ, τ ∈ N + be as in (5.1).
Define σn and τn for n = 0, 1, . . . by putting σ0 = σ and τ0 = τ , and iterating
(5.2)
σn+1 = σn + an+1 τn ,
zτn+1 = τn + ān+1 σn ,
where an are the Verblunsky coefficients of µ. This may be regarded as a bridge
between the Szegö recurrence and the Schur algorithm, in the following sense.
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
13
Proposition 5.3. Let µ be a probability measure with log w ∈ L1 , and let f be its
Schur function. Then the monic OPUC Φn of µ satisfy
hΦn = z n (h/h̄)σn + zτn
(5.3)
a.e.,
and the Schur iterates fn of f enjoy
(5.4)
fn (z) = τn (z)/σn (z)
(z ∈ D).
+
In particular, both σn and τn belong to N , and σn are outer.
Proof. The above iteration gives
z n+1 h̄−1 σn+1 + h−1 zτn+1
=z[z n h̄−1 σn + h−1 zτn ] + an+1 z n [z n h̄−1 σn + h−1 zτn ].
Since Proposition 5.1 says that h̄−1 σ + h̄−1 zτ = 1, this is nothing but the Szegö
recurrence formula (1.1). That is to say, (5.3) holds for every n = 0, 1, 2, . . . . By
Geronimus’ theorem, namely, an+1 = −ᾱn , the iteration also yields
τn+1 /σn+1 =
1 (τn /σn ) − αn
,
z 1 − ᾱn (τn /σn )
σn+1 = σn [1 − ᾱn (τn /σn )].
The former takes the same form as Schur’s iteration (4.3), while the latter implies
that σn+1 is outer if σn is outer and if τn /σn is a Schur function. Note that τn+1
belongs to N + if fn = τn /σn , because τn (0) + ān+1 σn (0) = 0 guarantees that
τn + ān+1 σn is divisible by z in N + . Since σ, τ ∈ N + , σ is outer, and f = τ /σ as
seen in the proof of Proposition 5.1, the remaining part is verified by induction. ¤
Qn
Remark 5.4. The above iteration gives |σn |2 − |τn |2 = j=1 (1 − |aj |2 ). Naturally,
one may also consider the ‘normalized’ pairs with |sn |2 − |tn |2 = 1, obeying
sn+1 = ρ−2
ztn+1 = ρ−2
n+1 (sn + an+1 τn ),
n+1 (tn + ān+1 sn ),
p
where ρn = 1 − |an |2 . These in turn take care of the orthonormal polynomials
ϕn of µ as hϕn = z n (h/h̄)sn + ztn . Since fn = tn /sn , Proposition 5.1 shows that
hn = 1/(sn − ztn ) is the Szegö function of a probability measure µn with fn .
The following provides a trick in the proofs of the main results of this section.
Lemma 5.5. Let X be a nontrivial linear subspace of N + such that
{zg | g ∈ X} = {g ∈ X | g(0) = 0}.
If σ, τ ∈ X, then σn , τn ∈ X for every n = 0, 1, 2, . . ., and conversely, if σn , τn ∈ X
for some n = 0, 1, 2, . . ., then σ, τ ∈ X.
Proof. Use Proposition 5.3, the iteration (5.2) and its inverse version
σn−1 = ρ−2
n [σn − an zτn ],
p
where ρn = 1 − |an |2 .
τn−1 = ρ−2
n [zτn − ān σn ],
¤
2
Remark 5.6. In the space L2 , the orthogonal complement of z n (h/h̄)H−
∩ H 2 is the
n
2
2
2
closure of z (h/h̄)H + H− . The case X = H is of special interest in connection
with the expansion formula in Proposition 3.5, modified as
∞
X
(z n (h/h̄)[Hn∗ Hn ]k 1 − Hn [Hn∗ Hn ]k 1).
hΦn = h(0)
k=0
14
Y. KASAHARA AND N. H. BINGHAM
P∞
It is not difficult to see that k=1 [Hn∗ Hn ]k 1 converges in H 2 if and only if σ, τ ∈ H 2 .
In this case, the above modified formula reduces to (5.3) with
∞
∞
X
X
σn = h(0)
[Hn∗ Hn ]k 1,
τn = −h(0)
[Hn∗ Hn ]k Hn∗ z̄,
k=0
k=0
which solve (I − Hn∗ Hn )σn = h(0) and (I − Hn∗ Hn )τn = −h(0)Hn∗ z̄.
Here is the strong Szegö theorem [Sz2] (more precisely, Ibragimov’s version [I],
supplemented with a theorem of Golinskii–Ibragimov [GI]; for textbook treatment,
see [Si2, Chapter 6]) in terms of a and γ.
Theorem 5.7. For a nontrivial probability measure µ, the following are equivalent:
∞
∞
X
X
(i)
j|aj |2 < ∞;
(ii) µ satisfies (1.2) and
j|γj |2 < ∞.
j=1
R
j=1
−ijθ
Proof. Write Lj (w) = T e
log w dm. First, assume (i). Then, by the Golinskii–
P∞
Ibragimov theorem, µs = 0, log w ∈ L1 and j=−∞ |j||Lj (w)|2 < ∞, which implies
P∞
2
(ii) follows. Conversely,
j=1 |j||γj | < ∞ (see [Si2, Proposition 6.2.6]). Therefore,
P∞
assume (ii). Then it can be shown that kHn∗ Hn k ≤ j=n+1 j|γj |2 in the operator
norm on H 2 , by handling
(∞ Ã ∞
! )
∞
X
X X
γj+l γ̄k+l gk z j ,
(Hn∗ Hn g)(z) =
P∞
j=0
k=0
l=n+1
where g = j=0 gj z j ∈ H 2 . Hence, one may take n large enough so kHn∗ Hn k < 1,
P∞
making k=0 [Hn∗ Hn ]k 1 converge absolutely in H 2 . Since this gives σn , τn ∈ H 2 ,
Proposition 5.1 and Lemma 5.5 find that h−1 = σ − zτ ∈ H 2 . Therefore, w−1 ∈ L1 .
The results of Ibragimov–Solev [IS, Theorems 1 and 2] now implies w−1 = |p|2 v,
where p is a polynomials with roots on T and v is a nonnegative function such that
P
∞
2
∈ L1 (see [Si2, Proposition 6.1.5]). However,
j=−∞ |j||Lj (v)| < ∞, as well as v
P∞
1
since w ∈ L , p is constant, so that j=−∞ |j||Lj (w)|2 < ∞. Thus, by Ibragimov’s
theorem, (i) holds.
¤
The next is a similar extension for Baxter’s theorem [B] (see [Si2, Chapter 5] for
textbook treatment). Let ν be a Beurling weight, which is placed on Z in such a
way that νj ≥ 1, ν−j = νj and νj+k ≤ νj νk for all j, k ∈ Z. It is called a strong
Beurling weight if inf j∈Z (j −1 log νj ) = 0. For example, νj = (|j| + 1)α (j ∈ Z) is a
strong Beurling weight for α ≥ 0.
Theorem 5.8. Let ν be a strong Beurling weight. Then, for a nontrivial probability
measure µ, the following are equivalent:
∞
∞
X
X
(i)
νj |aj | < ∞;
(ii) µ satisfies (1.2) and
νj |γj | < ∞.
j=1
j=1
P∞
Proof. Write A for the Beurling algebra of L -functions g = j=−∞ gj eijθ with
P∞
kgkν = j=−∞ νj |gj | < ∞. Assume (i). Then µs = 0, log w ∈ L1 and h, h−1 ∈ A
(see [Si2, Theorem 5.2.2]). So, µ obeys (1.2) and
h̄ ∈ A, whence (ii). Conversely,
Ph/
∞
assume (ii). Then Hn∗ Hn satisfies kHn∗ Hn k ≤ ( j=n+1 νj |γj |)2 in A-norm. Hence,
in the same way as the above proof, one has σ, τ ∈ A. Thus, by Proposition 5.1,
h−1 ∈ A and h ∈ L∞ . These imply (i) (see [Si2, Theorem 5.2.2]).
¤
∞
VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES
15
References
[AAK] V. M. Adamjan,D. Z. Arov and M. G. Krein, Infinite Hankel matrices and generalized
problems of Carathéodory-Fejér and I. Schur, Funkcional. Anal. i Priložen. 2 (1968),
1–17; Functional Anal. Appl. 2 (1968), 269–281.
[B]
G. Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle,
Trans. Amer. Math. Soc. 99 (1961), 471–487.
[BIK] N. H. Bingham, A. Inoue and Y. Kasahara, An explicit representation of Verblunsky
coefficients, Statist. Probab. Lett., in press.
[BJH] P. Bloomfield, N. P. Jewell and E. Hayashi, Characterizations of completely nondeterministic stochastic processes, Pacific J. Math. 107 (1983), 307–317.
[D]
P. L. Duren, Theory of H p spaces, Academic Press, New York, 1970.
[DM] H. Dym and H. P. McKean, Gaussian processes, function theory, and the inverse spectral
problem, Academic Press, New York, 1976.
[G]
J. B. Garnett Bounded analytic functions. Academic Press, New York, 1981.
[GI]
B. L. Golinskii and I. A. Ibragimov, A limit theorem of G. Szegö, Izv. Akad. Nauk SSSR
Ser. Mat 35 (1971), 408–427, [Russian]
[H]
P. R. Halmos, A Hilbert space problem book. Springer-Verlag, New York, 1982.
[Ha]
E. Hayashi, The solution sets of extremal problems in H 1 , Proc. Amer. Math. Soc. 93
(1985), 690–696.
[Ho]
K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, 1962
[I]
I. A. Ibragimov, A theorem of Gabor Szegö, Mat. Zametki 3 (1968), 693–702 [Russian].
[IS]
I. A. Ibragimov and V. N. Solev, A condition for the regularity of a Gaussian stationary
process, Dokl. Akad. Nauk SSSR bf 185 (1969), 509–512 [Russian]. English transl. in Soviet
Math. Dokl. 10 (1969), 371–375.
[In1] A. Inoue, Asymptotics for the partial autocorrelation function of a stationary process,
J. Anal. Math. 81 (2000), 65–109.
[In2] A. Inoue, Asymptotic behavior for partial autocorrelation functions of fractional ARIMA
processes, Ann. Appl. Probab. 12 (2002), 1471–1491.
[In3] A. Inoue, AR and MA representation of partial autocorrelation functions, with applications, Probab. Theory Related Fields 140 (2008), 523-551.
[IK1] A. Inoue and Y. Kasahara, Partial autocorrelation functions of the fractional ARIMA
processes with negative degree of differencing, J. Multivariate Anal. 89 (2004), 135–147.
[IK2] A. Inoue and Y. Kasahara, Explicit representation of finite predictor coefficients and its
applications, Ann. Statist. 34 (2006), no. 2, 973–993.
[LM] N. Levinson and H. P. McKean, Weighted trigonometrical approximation on R1 with
application to the germ field of a stationary Gaussian noise. Acta Math. 112 (1964), 99–
143.
[dLR] K. de Leeuw and W. Rudin, Extreme points and extremum problems in H1 , Pacific
J. Math. 8 (1958), 467–485.
[N1]
T. Nakazi, Exposed points and extremal problems in H 1 , J. Funct. Anal. 53 (1983),
224-230.
[N2]
T. Nakazi, Kernels of Toeplitz operators, J. Math. Soc. Japan 38 (1986), 607–616.
[Ne]
Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153-162.
[P]
V. V. Peller, Hankel operators and their applications. Springer-Verlag, New York, 2003.
[PS]
A. Poltoratski and D. Sarason Aleksandrov-Clark measures, Recent advances in operatorrelated function theory, 1–14, Contemp. Math. 393, Amer. Math. Soc., Providence, RI,
2006.
[S1]
D. Sarason, Function theory on the unit circle, Notes for lectures given at a Conference
at Virginia Polytechnic Institute and State University, 1978.
[S2]
D. Sarason, Exposed points in H 1 , I, Oper. Theory Adv. Appl. 41 (1989), 485–496.
[S3]
D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, Wiley, 1994.
[Se]
A. Seghier, Prédiction d’un processus stationnaire du second ordre de covariance connue
sur un intervalle fini, Illinois J. Math. 22 (1978), no. 3, 389–401.
[Si1]
B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, American
Mathematical Society, Providence, RI, 2005.
[Si2]
B. Simon, Orthogonal polynomials on the unit circle, Part 2. Spectral theory, American
Mathematical Society, Providence, RI, 2005.
16
Y. KASAHARA AND N. H. BINGHAM
[Si3]
[Sz1]
[Sz2]
[V1]
[V2]
[Y]
[Yo]
B. Simon, Szegö’s theorem and its descendants. Spectral theory for L2 perturbations of
orthogonal polynomials, Princeton University Press, Princeton, NJ, 2011.
G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939
G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive
function. Comm. Sém. Math. Univ. Lund 1952 (1952), Tome Supplementaire, 228–238.
S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier
series, Proc. London Math. Soc. 38 (1935), 125–157.
S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc.
40 (1936), 290–320.
K. Yabuta, Some uniqueness theorems for H p (U n ) functions. Tôhoku Math. J. 24 (1972),
353–357.
R. Younis, Hankel operators and extremal problems in H 1 , Integral Equations Operator
Theory 9 (1986), 893–904.
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
E-mail address: [email protected]
Department of Mathematics, Imperial College London, London SW7 2AZ
E-mail address: [email protected]